An aircraft air conditioning system is used to set and maintain a desired pressure, a desired temperature and a desired air humidity in an aircraft cabin. Moreover, the aircraft air conditioning system supplies sufficient fresh air to the aircraft cabin to ensure that a prescribed minimum amount of fresh air is available in the aircraft cabin. An aircraft air conditioning system is known from EP 2 735 510 A1 and US 2014/0144163 A1 in which a refrigerating apparatus operated using a two-phase refrigerant is used to cool ambient air compressed by a multistage compressor. The refrigerating apparatus comprises a refrigerant circuit, arranged or disposed in which are a compressor, a liquefier, an expansion valve and an evaporator, through which ambient air that is to be cooled flows. Bleed air drawn from an engine or an auxiliary power unit of the aircraft is used to power the compressor of the refrigerating apparatus and to power the multistage compressor for compressing the ambient air.